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Full text of "Treatise On Analysis Vol-Ii"

21    PRODUCT OF MEASURES       239

Let 9? be the restriction of the canonical mapping TT : R2 -» T2 to the set K of
points (x, y) such that 0 <; x < I and 0 £y < 1, so that 9? is a continuous bijection of
K onto T2. For t e T2 and (x, y) « g?" J(0, put

p     =

\ir(2x, %(y + I))

Show that the mapping v : T2-»T2 ("baker's transformation") is continuous almost
everywhere and that v o/=/0 w (where u is the mapping defined in (a)).

We shall see later that u and v are ergodic (relative to v and /3, respectively)
(Section 15.11, Problem 16).

(c)   Show that for the Bernoulli scheme B(/?0, ...,/?* _i), the entropy /z(«) (Section
13.9, Problem 28) is given by

h(u) = -(po log A) H

(Use the KolmogorofT-Sinai theorem, by starting with the partition a =

where Aj is the set of (xn)H9Z such that XQ =j. Use Problem 9(a) and the definition

of «, and remark that the set

A,0 n u-^AjJ n • • - n u-'+^Aj^,)

is the set of (.*„)„ «z such that XQ =j0 , xi =yl5 . . . , x,,-! = ^-i. The problem is then to
calculate the sum

£                .      - -Pomo -

mo + • • • + n\ti - i ™ n /^Q ! " * " Wfc _ \ I

For this purpose, observe that

Z         — ~ - rWoPo0"1

mo + • • • + "'fc - 1 = " WQ I ' " ' MX, - i I                                                               up®

Deduce from this calculation that the bijections u of the Bernoulli schemes B(J, J)
and B(|, i, I) are not conjugate (Section 13.12, Problem 11).

19.   Let X be a locally compact space, let ^ be a positive measure on X, and let /, g be
two nonnegative /^-measurable functions on X. For each a > 0, let A« be the set of
x e X such that f(x) > a. Suppose that :
(i)   /x(A«)< -f oo for all a>0;
(ii)   g e ^£(X, /x) for some/7 e ]1, + oo[;

(iii)   for each a > 0, /x(Aa) ^ —

a J A

Show that these conditions imply that/e ^f(X, ft) and that

Np(/) g

- 1

(Wiener's inequality). (Consider first the case where / is bounded and of compact
support, and integrate from 0 to oo the inequality0 and ^pj = 1). On each factor